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Global Maximum and Global Minimum

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Consider the following function:

f(x,y) = xy
on the set S = {x^2 +4y^2 ≤ 1}.

a) Explain by applying a relevant theorem why f(x,y) has a global maximum and a global minimum in the set S.

b) Find the critical of f in the interior of the set S.

c) Use the method of Lagrange multipliers to find the minima and maxima of f on the boundary of S given by x^2 + 4y^2 =1.

d) Find the global maximum and the global minimum of f.

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Solution Preview

(a) The set S contains all the points inside and on the ellipse given by (see attached file for equations)
We take x = cos(theta); y = 1/2sin(theta) so that every point of the form P(cos(theta), 1/2sin(theta) satisfies the region given by S for 0 (less than or equal to) theta (less than) 2pi
Hence the function f(x, y) =xy can be written ...

Solution Summary

In the following posting, a function of two variables is studied for its global maximum and global minimum using two different methods including the Lagrange multiplier method. The questions are provided for the calculations. The data is provided for the relevant theorems.